SAT Math: Special Right Triangles (30-60-90 and 45-45-90)
29+ practice questions in Praczo
The concept, explained
- 1
45-45-90 triangle: sides are in ratio 1 : 1 : √2. If legs = x, hypotenuse = x√2.
- 2
30-60-90 triangle: sides are in ratio 1 : √3 : 2. Short leg = x, long leg = x√3, hypotenuse = 2x.
- 3
These ratios are on the SAT reference sheet — but use them fluently to save time.
- 4
To find a side, identify the triangle type, set the ratio, and solve for x.
- 5
These appear often in circles (inscribed angles, radii to tangent lines) and square/equilateral triangle problems.
- ✗ In a 30-60-90 triangle, mixing up which leg is short (opposite 30°) and which is long (opposite 60°).
- ✗ Forgetting to divide by √2 when given the hypotenuse of a 45-45-90 and solving for the legs — rationalize the denominator.
SAT-style practice
In a 30-60-90 triangle, the hypotenuse is 10. What is the length of the shorter leg?
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